Platonic Relations and Mathematical Explanations∗

Robert Knowles

8th September 2020

Abstract

Some scientific explanations appear to turn on pure mathematical claims.The enhanced indispensability argument appeals to these ‘mathematical ex-planations’ in support of mathematical platonism. I argue that the success ofthis argument rests on the claim that mathematical explanations locate puremathematical facts on which their physical explananda depend, and that anyaccount of mathematical explanation that supports this claim fails to providean adequate understanding of mathematical explanation.

Words: 8,142 (not including bibliography).

1. Introduction

Some scientific explanations appear to turn on pure mathematical claims. Accord-

ing to an influential line of argument, these mathematical explanations evidence

mathematical platonism—the view that abstract mathematical objects exist. The

argument can be simply stated as follows. We should infer to our best scientific ex-

planations. Some of these turn on pure mathematical claims, ascribing properties

to abstract mathematical objects. So, we should believe in abstract mathematical

objects. This is the enhanced indispensability argument (EIA).1

∗The final version of this paper is forthcoming in The Philosophical Quarterly, doi: ht-tps://doi.org/10.1093/pq/pqaa053.1See Baker 2005. EIA ‘enhances’ the traditional indispensability argument, attributed to Quine 1948and Putnam 1971, by doing without the controversial principle that confirmation is radically holistic.See Sober 1993, Maddy 2005, and Morrison 2012 for relevant criticism of the traditional argument.

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EIA supporters take mathematical explanations to work by locating explanat-

ory mathematical facts.2 Critics take them to work by exploiting mathematics as

a representational aid to pick out otherwise elusive explanatory physical facts.3

Progress in this debate requires a mature enough understanding of mathematical

explanation to adjudicate between these views.

In recent years, a number of philosophers have defended accounts of math-

ematical explanation that appear promising for bolstering EIA: Christopher Pin-

cock argues that certain explanations locate mathematical facts on which their ex-

plananda depend, via a sui generis, non-causal dependence relation; Sam Baron,

Mark Colyvan, David Ripley, and Mark Povich suggest that mathematical explan-

ations locate the mathematical facts on which their explananda counterfactually

depend; and Aidan Lyon argues that mathematical explanations identify causally

relevant mathematical properties.4 These accounts appear promising for bolstering

EIA because they are ontic: according to them, mathematical explanations limn

the network of objective dependencies in which their explananda are embedded,

and locate mathematical facts within this network.5 In contrast, relatively few have

defended accounts that seem promising for undermining EIA.6 It may therefore

appear that the balance of evidence currently favours EIA. I aim to dispel this ap-

pearance by providing support for the following argument.2E.g. Colyvan 2002, 2013; Baker 2005, 2009, 2017; Lyon 2012.3E.g. Daly & Langford 2009; Leng 2010, 2012; Rizza 2011; Saatsi 2011; Yablo 2012.4Baron 2020; Baron, Colyvan & Ripley 2017; Lyon 2012; Pincock 2015b; Povich 2019.5I assume that there are non-causal relations of explanatory dependence, in accordance with somedominant thinking about grounding (see Schaffer 2009; Rosen 2010; Audi 2012; Clark & Liggins2012; Raven 2015). I help myself to, and use interchangeably, the associated locutions ‘in virtueof’, ‘depends on’, and ‘is grounded by’. Two further clarifications: First, I talk as though groundingcan relate many different kinds of things; if it turns out to be more selective, my arguments can beregimented accordingly. Second, I do not presume grounding is unitary; for all I say, it may be anon-committal means of discussing a range of more specific relations (see Wilson 2014).

6See Knowles & Saatsi 2019 and Leng 2012.

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PREMISE 1

Any account on which mathematical explanations locate pure math-

ematical facts on which their physical explananda depend fails to provide

an adequate understanding of mathematical explanation.

PREMISE 2

Any account of mathematical explanation apt to bolster EIA implies

that mathematical explanations locate pure mathematical facts on which

their physical explananda depend.

CONCLUSION

Any account of mathematical explanation apt to bolster EIA fails to

provide an adequate understanding of mathematical explanation.

Part of the support I provide for this argument is piecemeal: I identify the three

above-mentioned accounts of mathematical explanation as confirming instances of

the generalisations PREMISE 1 and PREMISE 2 (§§2–4).

In §2, we see Pincock’s (2015b) account faces two serious problems. First, it

posits striking yet inexplicable regularities. This is the brute regularities problem.

Second, it renders our supposed success in identifying the mathematical facts on

which physical phenomena depend a cosmic coincidence. This is the empirical

access problem. These problems reveal that Pincock’s account fails to provide

an adequate understanding of mathematical explanation. It is thus a confirming

instance of PREMISE 1.

In §3, I show that, for the increasingly popular countermathematical account

to provide any understanding of mathematical explanation, it must posit a sui gen-

eris physical-on-mathematical dependence relation. The result is that it also faces

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the brute regularities and empirical access problems and is a further confirming

instance of PREMISE 1.

In §4, I show that Lyon’s (2012) account is best understood in terms of the

dependence of mixed mathematical/physical facts on physical facts. In contrast

to the other accounts, the direction of dependence is reversed here, which means

the brute regularities and empirical access problems are avoided. However, the

cost is the ability of Lyon’s account to bolster EIA. Lyon’s account is a confirming

instance of PREMISE 2.

The argumentative moves in each case are responsive to the same general reas-

ons. Capitalising on this, I will formulate a general argument for CONCLUSION,

in the form of a dilemma (§5). Finally, I will show that, despite initial appear-

ances, adopting structuralism about mathematical objects will not help to avoid

this dilemma (§6). All of this provides substantial (albeit defeasible) support for

CONCLUSION.

2. Abstract Dependence

Pincock (2015b) claims that certain scientific explanations account for the prop-

erties of collections of physical systems by appealing to the properties of objects

more abstract than those physical systems. He calls these abstract explanations, and

posits the sui generis, non-causal relation of abstract dependence to make sense of

them. His chosen case study is an explanation in which the more abstract objects

are mathematical.

If Pincock is right, then his account bolsters EIA. In this section, I show that

Pincock’s account faces the brute regularities and empirical access problems, and

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so fails to provide an adequate understanding of mathematical explanation. First, I

show that Pincock’s account leaves it underdetermined which mathematical fact is

selected for by the abstract dependence relation. Then I argue that we must never-

theless take instances of abstract dependence to select for particular mathematical

facts, as Pincock does. Finally, I show that this leads to the brute regularities and

empirical access problems.

Pincock’s case study is the mathematical proof that soap formations must sat-

isfy Plateau’s laws. Plateau’s laws capture three striking regularities in soap film

and bubble formations:

(1) Soap formations consist of finite flat or smoothly curved surfaces smoothly

joined together.

(2) Within a soap formation, there are three possible meetings of surfaces: (i)

no surfaces meet; (ii) exactly three surfaces meet along a smooth curve; (iii)

exactly six surfaces (together with four curves) meet at a vertex.

(3) When three surfaces meet along a curve, they do so at angles of 120◦; when

four curves meet at a point, they do so at angles of ≈ 109◦.

The proof, from Jean E. Taylor, can be divided into three parts (following Almgren

& Taylor 1976). The first is the initial modelling phase, where a mathematical ana-

logue of soap formations is defined, capturing the basic physical principle that soap

formations minimize their total surface area. For a soap formation on a wire frame,

the area-minimization leaves the frame’s size unchanged. For a bubble forma-

tion, the area-minimization leaves the volume of enclosed air unchanged. These

properties are captured by approximating soap formations with configurations of

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two-dimensional surfaces in R3 that are minimal: their total area cannot be de-

creased by certain small deformations that leave their frame or enclosed volume

fixed. These configurations are almost minimal sets.

The second part of the proof shows that almost minimal sets that satisfy (1)

also satisfy (2) and (3). The final part shows that almost minimal sets exist and

satisfy (1). Overall, the proof shows that almost minimal sets satisfy Plateau’s three

laws, which is relevant to soap formations because the defining property of almost

minimal sets models the area-minimization principle that governs soap formations.

Pincock (2015b) claims that this proof explains the fact that soap formations

satisfy Plateau’s laws by showing that it abstractly depends on the fact that al-

most minimal sets are minimal. For this to provide an adequate understanding of

mathematical explanation, we must have a decent understanding of the abstract de-

pendence relation. Pincock is clear that it is objective, non-causal, and sui generis,

but going beyond this very general characterization raises serious problems.

Abstract dependence is supposed to obtain between a fact about certain phys-

ical systems, and a fact about some more abstract objects. To elucidate this idea,

Pincock (2015b: 865–866) appeals to the relationship between types and tokens. A

piece of music (type) and a particular performance of it (token) share many proper-

ties. For example, assuming the performance is faithful and successful, they share

many of their aesthetic properties. Yet there are certain more specific properties

the token alone has, such as having a particular spatial location. In this way, the

piece of music is more abstract than the performance of it, and the latter is an

instance of the former. Similarly, we can say that something is an instance of an

almost minimal set just in case it has minimality, and other more specific properties

besides.

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So, a condition on physical-on-mathematical abstract dependence is that the

physical systems are instances of the mathematical objects, in the above sense.

This is not a sufficient condition. To see this, imagine an accurate plastic model

of a soap formation created by a teacher by combining plastic surfaces so as to

satisfy one of Plateau’s laws. This is an instance of an almost minimal set, but

the model satisfies Plateau’s laws because the teacher made it so. According to

Pincock (2015b: 866–867), for abstract dependence to obtain, the formation of the

physical systems must be governed by a process relevant to the condition for being

instances of the more abstract objects. The relevant fact about soap formations is

eligible because soap formations are instances of almost minimal sets, and because

they are formed by a process of area-minimization that is relevant to their being

instances of almost minimal sets.

This gives us an idea of what abstract dependence requires on the physical side.

But what about the mathematical side? We know that the mathematical objects

must have the physical systems as instances, but this fails to distinguish between

many distinct candidates. There are at least three sources of underdetermination.7

7Pincock (2015b: 877) admits that it is possible to define distinct kinds of mathematical objects andrelate them to soap formations in ways that mimic Taylor’s proof. Perhaps he has in mind oneor more of the sources of underdetermination I identify, or perhaps he has in mind some furthersource. In any case, he recognizes this as a problem, and suggests that the remedy is a theory aboutwhat abstract dependence is and how it is distributed. If my discussion in this section is right, noameliorative theoretical moves are forthcoming. In another paper (2015a), where Pincock applieshis account to explanations within mathematics, he identifies an analogue of the same problem.There he proposes that the abstract ground for a given explanandum is the least more abstract factamong the candidates (2015a: 12). This solution presupposes that the various candidates for abstractgrounds are partially ordered by their abstractness. It is far from clear that they are, at least for thecandidates I identify. However, even if they are, and there is a least more abstract fact in the offing,why choose this, as opposed to, say, the most more abstract fact? This solution seems undulyarbitrary.

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Properties: There are various definitions of ‘minimality’ in terms of which the

explanation can be run. For example, Guy David (2013) provides a variation on

the definition offered by F. J. Almgren Jr. (1975), invoking different sets to fix

the operative notion of a small deformation. David says ‘a few other variants ex-

ist, but they would not be significantly different for what we want to say here’

(2013: 77), where among what he wants to ‘say’ is Taylor’s proof. Even if these

alternatives are extensionally equivalent, explanation is more fine-grained than ex-

tensional equivalence. Abstract dependence is supposed to select for a particular

fact involving certain mathematical objects having certain properties. So, there are

at least as many candidates as there are distinct properties in terms of which the

explanation can be run.

Bearers: For a given property, we face the further choice of bearers. For ex-

ample, we could restrict our attention to the almost minimal sets contained within

a particular subregion of R3, such as the interior of a particular sphere, with no

overall effect on the explanation. There are at least as many further candidates for

abstract grounds as there are distinct collections of bearers in terms of which we

can run the explanation.

Interpretations: Given a selection of properties and bearers, we face a further

choice of interpretation. For example, there are many distinct set-theoretic models

of R3, each of which has its own collection of almost minimal sets. The mathem-

atics of almost minimal sets (topology and geometric measure theory) is algebraic,

meaning it has no intended model. In light of this, no particular model of R3 has

claim to being the more natural home for the explanation. So, there are at least as

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many further candidates for the abstract grounds as there are models of R3.

These three sources of underdetermination are independent and cross-cutting. In

light of them, our understanding of the abstract dependence relation seems inad-

equate. By ruling out the alternative options, I will now argue that the only sensible

recourse is to claim that instances of abstract dependence select for a particular

mathematical fact from among the eligible candidates.

As I see it, there are three alternative options. The first is to argue that each

candidate abstract ground is partial, so that only by enumerating every candidate

have we provided the full abstract explanation. On this view, Taylor’s explanation

is incomplete. This cannot be right. Once we have understood the explanation, we

stand to learn nothing of explanatory value by considering other candidate abstract

grounds. Moreover, a philosophical theory should not pass judgement on the suc-

cess of a scientific explanation. If practitioners deem it successful, a naturalistic

philosophical account should take its success as a datum. There is also the further

difficulty that, on this view, the explanation looks to be impossible to complete.

The second option is to argue that each candidate abstract ground is complete.

On this view, each instance of abstract dependence is a case of massive overde-

termination of a particularly problematic kind. To illustrate, contrast the present

case with the much-discussed case of two people simultaneously throwing a rock

through a pane glass window. Each rock-throw is sufficient to cause the glass to

break all on its own, but both rocks hit the window at the same time, so both events

cause the window to break. This is a case of causal overdetermination, but it is

unproblematic for at least two reasons.

The first is that its occurrence does not involve systematic coincidence. Even if

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it is a coincidence that the rocks were thrown at exactly the same time, or that the

rocks hit the window at the same time, one-off coincidences of this kind shouldn’t

concern us. The second is that, despite the overdetermination, the explanation of

why the window breaks is ‘causally satisfying: there is a precise account of the

causal powers of both rocks, and of the individual contribution of each rock to the

shattering of the window. Removing one rock-throw has an easily definable result:

the window shatters with less force’ (Bernstein 2016: 30).

In contrast, on the present proposal, every case of abstract explanation will in-

volve massive and systematic overdetermination. Further, the explanation is not

satisfying in above sense. We have no sense of what each abstract ground is con-

tributing, nor how the explanandum would change were any one of them to be

removed.

A further worry is that, if all it takes for something more abstract than the ex-

planandum to be a complete abstract ground is that it satisfies one of a range of

very general structural properties, then abstract grounds come too cheaply. This

is at odds with the ontic aspirations of Pincock’s account. Objective dependence

relations should demand a lot of their relata. Causation demands physical or modal

connections between its relata, and grounding demands more intimate metaphys-

ical connections. On the present proposal, abstract dependence merely requires

certain kinds of similarity. It is hard to understand how this would amount to ob-

jective dependence in any particular direction. Moreover, if abstract dependence

comes too easily, we would expect it to crop up everywhere. As Pincock says,

‘[u]nless there is some principled way to constrain the proliferation of abstract de-

pendence relations, there will be too many of them and so the value of abstract

explanations will be diluted’ (2015b: 877).

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The final option is to argue that it is indeterminate which mathematical fact is

selected. To say that Taylor’s proof explains why soap formations satisfy Plateau’s

laws by identifying something on which the explanandum does not determinately

depend on, but also does not determinately not depend on, does not help us un-

derstand how this mathematical explanation works. This is surely a caricature of a

range of worked-out views that may provide some understanding; but the onus is

on the proponent of Pincock’s account to provide such a worked-out view. Without

one, tethering our understanding of mathematical explanation to our understanding

of how ontic indeterminacy interacts with ontic dependence seems like a bad way

to go.

In light of the inadequacy of the above three options, we are forced to conclude

that an instance of abstract dependence selects for a particular mathematical fact

from among the eligible candidates. But this incurs an explanatory debt. If abstract

dependence selects for one among a range of eligible mathematical candidates, it

seems there should be a reason why.8 We have seen that the conditions abstract

dependence imposes on its relata do not account for this. I can think of two further

ways of gaining understanding of a relation. First, we might identify symptoms of

its obtaining. Pincock suggests that the novelty and informativeness of the char-

acterization of soap formations as almost minimal sets may be symptomatic of

abstract dependence (2015b: 877–878). However, explanations run in terms of any

of the eligible candidates for the abstract grounds would be novel and informative

in these ways, so these symptoms fail to address the present concern. Indeed, I can

think of no symptoms that would.8Compare Benacerraf: ‘If the numbers constitute one particular set of sets, and not another, thenthere must be arguments to indicate which’ (1965: 58).

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Second, we might provide an analysis of the relation in more familiar terms.

Abstract dependence is sui generis, which rules out a reductive definition; but we

may be able to provide an illuminating non-reductive characterization. Pincock’s

invoking of the instantiation relation, along with his description of abstract depend-

ence as non-causal and objective is as close as he gets to such a characterization,

and we have already seen that these descriptors are far from illuminating.

At this point, it may be tempting to say that there is something which accounts

for why abstract dependence selects for the mathematical relata it does, though

we may never be in a position to know it. This last draw amounts to a desperate

assurance that the explanatory debt is settled, in spite of the lack of any reason

to think so. An account on which we may never properly understand the operat-

ive dependence relation is a poor foundation on which to build an understanding of

mathematical explanation. It seems our only recourse is to stipulate that, as a matter

of brute fact, abstract dependence selects for one among the many candidate ab-

stract grounds. This brings abstract dependence within reach of our understanding,

to the extent that it places no important features of it beyond our ken. Ultimately,

however, our understanding is no better off.

For a given instance of abstract dependence underlying a mathematical explan-

ation, there will be a brute fact of the matter about which mathematical fact it se-

lects for. Stipulating that there is nothing to explain here does not dispel the feeling

that there is. It is a striking regularity that soap formations satisfy Plateau’s laws

by virtue of one mathematical fact, rather than any of the other eligible candidates.

But, in taking it as brute, we relinquish any means by which we might illumin-

ate it. The same goes for each putative instance of abstract explanation. Thus,

adopting Pincock’s position involves positing a range of striking yet inexplicable

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regularities. This is the brute regularities problem.

Let us assume that Taylor’s proof succeeds in picking out the unique abstract

ground of our explanandum. That is, it locates the right properties of the right

mathematical objects. The proof is couched in terms of these properties of these

objects in the first place because they provide good approximate models of soap

formations, by bearing certain structural similarities to them. These structural sim-

ilarities must therefore have been a reliable guide to which mathematical fact the

explanandum abstractly depends on. But we have been forced to accept that ab-

stract dependence selects for abstract grounds as a matter of brute fact. Ipso facto,

it does not select for abstract grounds in virtue of any structural similarity they

bear to the physical systems whose properties they determine. But then Taylor’s

success in identifying the abstract ground of the fact that soap formations satisfy

Plateau’s laws, guided as it was by structural similarities, is a fluke. More generally,

assuming there are supposed to be many more cases of abstract explanation, Pin-

cock’s account implies that the reliability with which practitioners identify abstract

grounds is a massive cosmic coincidence. This is the empirical access problem.9

Because it faces these problems, Pincock’s account fails to provide adequate

understanding of mathematical explanation. The brute regularities problem shows

that it offers no understanding of how abstract grounds are related to what they

ground. The empirical access problem shows that it offers no understanding of

how practitioners succeed in providing abstract explanations. There are mysteries

precisely where an account of mathematical explanation should illuminate. Note

that my arguments here are sensitive only to the fact that Pincock’s view implies9This objection is reminiscent of Field’s (1989: 25–30, 230–239) variation on the epistemologicalobjection to mathematical platonism, which arguably improves on Benacerraf’s (1973).

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that physical phenomena bear an objective, sui generis dependence relation to pure

mathematical facts. The peculiarities of Pincock’s account are irrelevant. If other

accounts of mathematical explanation imply the same, we should expect them to

face the brute regularities and empirical access problems. Pincock’s account is a

confirming instance of PREMISE 1 for entirely general reasons.

3. Counterfactual Dependence

The notion that scientific explanation involves tracing relations of counterfactual

dependence is growing in popularity, due in no small part to James Woodward

and Christopher Hitchcock’s development of an influential counterfactual analysis

of causal explanation in science.10 Many authors have since argued that we can

generalize Woodward and Hitchcock’s account in pursuit of a monist counterfac-

tual theory that covers causal and non-causal explanations alike.11 In this spirit,

Baron, Colyvan, Ripley, and Povich offer counterfactual accounts of mathematical

explanation.12

I am sympathetic to this movement. However, there is more than one way

of extending the counterfactual theory to mathematical explanation. One might

take the mathematics in mathematical explanations to identify mathematical facts

on which their explananda counterfactually depend; or one might take it to help

identify non-mathematical facts on which their explananda counterfactually de-

pend. The aforementioned authors all develop the former option.13 This counter-

mathematical account seems promising for bolstering EIA. However, I will argue10Woodward 2003; Hitchcock & Woodward 2003; Woodward & Hitchcock 2003.11Saatsi & Pexton 2013; Rice 2015; Reutlinger 2016; Povich 2018.12Baron, et al. 2017; Baron 2020; Povich 2019.13Knowles & Saatsi 2019 develop the latter.

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that its proponents must posit and evidence the existence of a sui generis relation

of non-causal dependence running from the physical to the mathematical. Because

of this, for all the reasons detailed in §2, the countermathematical account faces

the brute regularities and empirical access problems, and thus fails to provide an

adequate understanding of mathematical explanation. In other words, it is a further

confirming instance of PREMISE 1.

Baron et al. (2017) and Baron (2020) use the following case study.14 North-

American periodical cicadas lie dormant in larval form for either 13 or 17 years,

then emerge to eat, mate, and die. Why 13 and 17, specifically? Background

ecological constraints restrict cicadas’ life-cycles to between 12 and 18 years.

Within that range, periods that maximize the time between co-emergence with

nearby periodical predators will be advantageous. The number of years between

the co-emergence of two periodical organisms is equal to the least common mul-

tiple (LCM) of their life-cycles in years, and this is maximized when these num-

bers are coprime. There are good reasons for thinking that any nearby predators

will have life-cycle periods of less than 12 years, and prime numbers are coprime

with all numbers smaller than themselves. We therefore expect there to have been

evolutionary pressure for cicadas to evolve prime-numbered life-cycle periods, and

13- and 17-year periods, specifically.

On the countermathematical account, the above explanation works by identify-

ing the mathematical facts on which the explanandum counterfactually depends. It

does this by implicating specific countermathematical claims, such as the follow-

ing:14Introduced to the literature by Baker 2005.

15

CM If 13 and 17 were not prime, cicadas would not have 13- or 17-

year life-cycles.

To make sense of this, we at least need a procedure for evaluating countermathem-

aticals like CM . Baron et al. (2017) provide a two-step procedure relied upon by

each of the aforementioned proponents of the countermathematical account. We

first imagine a scenario in which the antecedent obtains, keeping the rest of the

scenario as similar to actuality as possible without giving rise to a contradiction

in the neighbourhood. In other words, we change just enough to produce a frag-

ment of mathematics in which the antecedent obtains consistently. This step is the

twiddle. We then rely on relevant physical laws to ascertain how the mathematical

changes ramify in the physical system. This step is the ramification. For example,

for CM , we imagine a scenario in which the multiplication function is different,

such that 2 and 6 are factors of 13, making any further changes required to produce

a consistent fragment of arithmetic (twiddle). Finally, we appeal to the laws of

evolutionary theory to see how this change ramifies (ramification).

On the countermathematical account, countermathematicals, such asCM , must

come out as true via the above procedure. This requires the assumption that the

twiddle not only changes the mathematical facts, but also changes certain phys-

ical facts along with it. Only then will the physical laws yield the truth of the

consequent. This is a substantive assumption, and requires the positing of a rela-

tion between the mathematical and physical facts. Recognising this, Baron et al.

(2017: 9) suggest that structure-preserving mappings (homomorphisms) may suf-

fice. They won’t. There is a homomorphism between my fingers and my toes, but

that doesn’t give us any reason to think that, if I had nine fingers, I would have nine

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toes! The point generalizes. Homomorphisms are relations of similarity, grounded

in the properties of their relata. Changing the properties of one relatum only serves

to break the similarity; it does not force the other relatum to change along with it.

One might reply as follows. When we evaluate a counterfactual of any kind,

we have to make decisions about what to hold fixed and what to vary. All the above

objection shows is that we haven’t held enough fixed. If we hold the homomorph-

ism between the physical and the mathematical fixed, then the twiddle will ramify

in the desired way.15 This move is methodologically suspect. To see why, it will

be helpful to consider the evaluation of more humdrum counterfactuals. Suppose

I almost drop a very fragile cup on my tiled kitchen floor, but only just manage to

catch it. The following counterfactual is plausible: If I had not caught the cup, it

would have smashed. Following the standard procedure, we evaluate this counter-

factual by imagining a situation as similar as possible to actuality, but where the

antecedent is true. In this situation, it seems, I drop the cup, and the cup smashes.

One might think that, in the course of my evaluation, I have decided to hold

fixed a range of facts: the fact that the floor is tiled, the fact that the cup is fragile,

the fact that a very soft rug didn’t suddenly appear on the floor, the fact that an

angel wasn’t waiting to catch the cup in my place, and so on ad nauseam. But this

is to misunderstand the procedure. It is not a case of deciding what to hold fixed to

get the truth-value we want; it is a matter of trying to work out what would remain

unchanged, if the antecedent were true. I left the above-mentioned facts unaltered

because, based on what we know about the world, there is good reason to believe

that my failing to catch the cup would not change these things.

In contrast, we have seen that, if the mathematical and physical domains are15This is exactly what Baron (2020: 549–542) and Baron et al. 2017: 9–10 suggest.

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related by mere homomorphism, there is good reason to think that the homomorph-

ism would not survive certain changes to the mathematical domain. To respond to

this by suggesting we hold the homomorphism fixed seems completely ad hoc.

Worse, it is hard to see how countermathematicals whose truth is guaranteed in

this way could bear any explanatory weight. We might just as well explain the

redness of my socks by claiming that it counterfactually depends on the redness of

my shirt. After all (holding their sameness in colour fixed), if my shirt were not

red, my socks would not be, either.

The above shows that we need a stronger relation to underpin the truth of the

relevant countermathematicals. In particular, we must posit a relation of physical-

on-mathematical dependence, and evidence its existence. If we have reason to

think such a relation obtains, we have reason to think that changes in the math-

ematical facts will ramify as desired. The dependence relation cannot be one that

obtains too easily. For instance, if the primality of 13 and 17 determines the prop-

erties of any homomorphic physical system, then changes in these properties will

have widespread ramifications, and there will be no interesting counterfactual con-

nection between the primality of 13 and 17 and the cicadas’ life-cycle periods. The

dependence must be more demanding, such that changes in the relevant properties

of 13 and 17 only ramify in changes to the cicadas’ life-cycles.16

There is no off-the-shelf dependence relation one can appeal to here: the remit

is far too specific. So, the proponent of the countermathematical account must posit

a sui generis, non-causal dependence relation, the existence of which is presum-

ably evidenced by the success of the explanations amenable to analysis in terms of

it. We saw in §2 that this path leads to the brute regularities and empirical access16Baron et al. (2017: 9–10; fn. 10) recognise this, which is perhaps why they opt for homomorphism.

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problems. Just as in the soap formation case, there are many candidate mathem-

atical facts eligible for the role of determining the cicadas’ life-cycle durations.

Again, there appear to be at least three sources of underdetermination.

Properties: The familiar definition of primality can be stated as follows. For

a ∈ Z+ where a > 1, a is prime iff, for any b, c ∈ Z+, bc = a only if b = 1 or

c = 1. However, there is an alternative definition that runs as follows, where x|y

means x is a factor of y. For a ∈ Z+ where a > 1, a is prime iff, for any b, c ∈ Z+,

a|bc only if a|b or a|c. In fact, these definitions pick out distinct properties. The

first defines a special case of irreducibility, while the second defines a special case

of primality. For the positive integers, the properties coincide, but in more abstract

algebraic structures they come apart. So, that 13 and 17 are irreducible and that

13 and 17 are prime are distinct facts, both of which imply the relevant facts about

LCM-maximization. The explanation runs equally well by appeal to either.

Bearers: For a chosen property, we face a further choice of bearers. For example,

by measuring life-cycles in months, we uniformly multiply by 12. Such a uniform

translation will preserve the LCM-maximizing structure, so we can just as well

take the explanation to work by implicating a countermathematical such as the

following: If, in addition to 12(1) and 12(13), 12(13) had factors 12(2) and 12(6),

then cicadas would not have 12(13)-month life-cycle periods.

Interpretations: There are infinitely many set-theoretic models of the positive

integers, and it makes no difference whatsoever to the explanation if we interpret

the mathematics as about one or other of these models.

19

For the same reasons outlined in §2, this radical underdetermination forces us to

stipulate that, as a matter of brute fact, the dependence relation in question selects

for a particular mathematical fact among the many eligible candidates. This posits

a striking yet inexplicable regularity, and there will be further striking yet inex-

plicable regularities associated with each mathematical explanation amenable to

the countermathematical account. Thus, the countermathematical account posits a

range of striking yet inexplicable regularities.

On the countermathematical account, the cicadas explanation works by identi-

fying the mathematical facts on which the explanandum counterfactually depends.

The reason 13 and 17 are appealed to in the first place is that they help to form

an adequate model of the physical system. The considerations at play in the initial

modelling must therefore be a reliable guide to what the explanandum counterfac-

tually depends on. These considerations include our desire to capture the hypo-

thesised structural relationship between the cicadas’ life-cycles and the life-cycles

of nearby predators, namely the minimization of co-emergence. Perhaps we also

consider the year to be a biologically significant unit of time, which may influence

our choice in units (Baker & Colyvan 2011: 329).

However, the counterfactual dependence in question is supported by a sui gen-

eris dependence relation that selects for its mathematical relata as a matter of brute

fact, meaning the natural numbers 13 and 17 are not selected in virtue of their

ability to capture structural features of the physical system, nor in virtue of the

privileged status of the year. Biologists’ success in identifying the mathematical

facts on which the cicadas’ life-cycle periods counterfactually depend is therefore a

massive coincidence. The countermathematical account faces the brute regularities

and empirical access problems, and so fails to provide an adequate understanding

20

of mathematical explanation. It is a further confirming instance of PREMISE 1.

4. Causal Relevance

Lyon (2012) analyses mathematical explanations in terms of Frank Jackson and

Philip Pettit’s (1990) theory of program explanation, with the explicit intention of

supporting EIA. To illustrate program explanation, imagine that water in a sealed

glass container is heated to boiling point. Why does the glass shatter? There are

two kinds of explanation. We can gesture towards the underlying causal process,

culminating in particular water molecules striking the glass with momenta collect-

ively sufficient to break it. This is the process explanation. Or we can point to

the water’s being at 100◦C. This is the program explanation. Jackson and Pettit’s

theory aims to show how these two explanations are related.

The relation is a modal one. The temperature property had to be realized by

some arrangement of molecules and some distribution of momenta among them,

such that some molecules or other would have struck the glass with momenta col-

lectively sufficient to break it. Because we know the temperature property was

instantiated by the water, we can be sure that some causal process or other pro-

duced the explanandum. We say that the temperature property programs for the

causes of the explanandum, and is thereby causally relevant to it.

Why do we need program explanations, if we are so sure there are process

explanations in the offing? The reason is twofold. First, underlying causal pro-

cesses are often recondite, so it is beneficial to have a means of exploiting their

existence for explanatory ends without having to describe them explicitly. Second,

program explanations yield explanatory information that process explanations do

21

not. Even if we were able to trace the trajectory of the molecules that shattered the

glass, doing so would miss the fact that, even if these particular molecules were not

responsible, some other molecules would have been. The program explanation im-

proves on the process explanation by implying that, whatever the underlying causal

goings on, so long as the temperature property was instantiated, some causal pro-

cess or other will have culminated in the glass shattering. In this way, program

explanations reveal the modal robustness of their explananda.

Lyon claims mathematical explanations work by locating a causally relevant

property of mathematical objects. For example, in the cicadas case, the instanti-

ation of being prime by 13 and 17 is supposed to program for the causes of the

explanandum. Unfortunately, Lyon fails to explain how this might work. Nor does

any promising account seem forthcoming (Saatsi 2012). Such an account requires

appeal to a dependence relation to supply the requisite modal force. In the temper-

ature example, we appeal to realization: we say that the temperature property had

to be realized by something sufficient for the relevant causes to be instantiated. But

the fact that 13 and 17 are prime does not obviously demand anything of cicadas,

in the way that the temperature property demands something of the water.

The temperature case suggests that programming occurs internal to the phys-

ical system, from a higher-level property instantiated by it, to lower-level causes

instantiated in it. If we are willing to give up on the contention that pure mathem-

atical properties program for causes, there is a natural way of getting the primality

of 13 and 17 involved with programming. There is a homomorphism between the

physical system and the mathematical domain that preserves the minimization of

the frequency of co-emergence as the LCM-maximization of 13 and 17 with in-

tegers smaller than 12. This mapping is determined by our decision to measure

22

life-cycle periods in years, along with the other background constraints imposed

by the explanation. Call this mapping φ. In virtue of it, the cicadas’ life-cycle

periods instantiate the higher-level, multiply-realizable properties being mappedφ

to 13 and being mappedφ to 17.

These properties tick all the boxes. Any period that instantiates one will be

part of a system featuring a minimization of the frequency of overlap between it

and shorter periods. Some causes or other will have led to this, so the instantiation

of one of these properties programs for the causes of the explanandum. In partic-

ular, its stable instantiation within a biological system programs for evolutionary

pressure towards that stability. Moreover, the existence of the positive integers is

required to enter into the mapping on which the instantiation of these properties

depends. So we have a metaphysic that explains how mathematical properties can

be causally relevant to physical phenomena, and appears promising for supporting

EIA.

This metaphysic does not entail that physical facts depend on mathematical

facts. The instantiation of the impure mathematical programming properties de-

pends on the overlap-minimizing structure of the physical system and the LCM-

maximizing structure of the mathematical domain, respectively. Because of this, it

is not a problem that there are many distinct mathematical candidates for capturing

the relevant properties of the physical system. We can happily say that the phys-

ical system instantiates a distinct programming property for each candidate, since

this will not result in any troubling systematic overdetermination. (Compare: the

fact that the water in the temperature example has a distinct programming property

corresponding to each temperature measurement scale does not over-determine the

explanandum.) We are not forced to posit any brute regularities, so our success

23

in identifying the programming properties is not mysterious. The present proposal

therefore appears to offer some understanding regarding how mathematical explan-

ations work.

Unfortunately, despite initial appearances, this account is unfit to support EIA.

In locating our causally relevant impure mathematical properties, we have inadvert-

ently located a purely physical property that can do all of the desired explanatory

work. We avoid the brute regularities and empirical access problems by accept-

ing that the instantiation of the programming properties is partially grounded by

the overlap-minimizing structure of the physical system. All the work done by the

impure mathematical property can be done by this overlap-minimizing property.

It is a multiply-realizable property of the physical system that programs for the

causes of the explanandum, but its instantiation does not require the existence of

any mathematical objects. Moreover, it is mutually-recognized among parties to

the debate. Indeed, the proponent of Lyon’s account needs it to ground the relevant

mapping.

The critic of EIA can achieve the same level of understanding of mathemat-

ical explanation by appeal to the same theory of explanation, while claiming that

the mathematics in mathematical explanations merely serves to represent a phys-

ical higher-level structural property of the physical system. Importantly, the failure

to support EIA is a consequence of relinquishing the claim that the physical ex-

plananda of mathematical explanations depend on pure mathematical facts. Lyon’s

account confirms PREMISE 2.

24

5. A Dilemma

Recall my master argument:

PREMISE 1

Any account on which mathematical explanations locate pure math-

ematical facts on which their physical explananda depend fails to provide

an adequate understanding of mathematical explanation.

PREMISE 2

Any account of mathematical explanation apt to bolster EIA implies

that mathematical explanations locate pure mathematical facts on which

their physical explananda depend.

CONCLUSION

Any account of mathematical explanation apt to bolster EIA fails to

provide an adequate understanding of mathematical explanation.

In §§2-4, I identified two confirming instances of PREMISE 1, and one confirming

instance of PREMISE 2. I showed that positing a sui generis relation of physical-on-

mathematical dependence fails to provide an adequate understanding of mathemat-

ical explanation (§2). I showed that the increasingly popular countermathematical

account must posit a sui generis relation of physical-on-mathematical dependence,

and so fails to provide an adequate understanding of mathematical explanation (§3).

And I showed that, while the causal relevance account seems able to provide some

understanding, it fails to support EIA precisely because it does not posit physical-

on-mathematical dependence (§4). These findings are significant in their own right.

25

But one might think they only provide piecemeal support for CONCLUSION. Not

so. My arguments are responsive to entirely general reasons. Capitalising on this,

we can give a general argument in favour of CONCLUSION in the form of a di-

lemma.

Any account of mathematical explanation apt for supporting EIA must be ontic.

It must characterise mathematical explanations as limning the network of object-

ive dependencies in which the explanandum is embedded, and locate explanatory

mathematical facts within this network.17 Such an account must choose between

two options. First, take the mathematical facts invoked by mathematical explan-

ations to depend on their physical explananda. Second, take their physical ex-

plananda to depend on the mathematical facts they invoke.18

Platonic mathematical facts do not depend on contingent physical facts, so

mounting the first horn with respect to a given explanation involves locating an

impure mathematical fact that depends on the physical explanandum via an inde-

pendently established relationship. If anything about a mathematical domain is

explanatory with respect to a physical phenomenon, it is its structure, and only

if the physical phenomenon is related to the mathematical domain via some kind

of mapping. So, our independently established relationship will be some kind of

mapping between the physical and the mathematical domain. However, mappings

obtain in virtue of the structures exhibited by their relata. As such, any explan-

atory relationship the mathematical structure bears to the physical explanandum

(via the mapping) will be mediated by the structure of the physical system, and the17I take this to be obvious and will not argue for it here. If I am wrong, then CONCLUSION must

be weakened as follows: Any ontic account of mathematical explanation apt to bolster EIA fails toprovide an adequate understanding mathematical explanation. This is still a significant result.

18I am omitting the complication that we can also choose whether the relevant dependence is partialor complete, which ultimately makes no difference to my argument.

26

latter will be just as eligible for bearing the proposed explanatory relationship to

the explanandum. So, there will be a mutually-recognized physical property that

does the required explanatory work. Mounting this horn results in an account of

mathematical explanation that cannot support EIA, whatever its other virtues.

Mounting the second horn with respect to a given explanation involves locating

some mathematical fact on which the physical explanandum depends. To provide

an adequate understanding of this dependence, the account will have to say some-

thing about how it selects for its mathematical relatum. Since mathematical objects

are abstract, this will be in terms of the mathematical objects’ fulfilment of a cer-

tain theoretical role, such as capturing structural features of the physical system.

However, for any specified theoretical role, there will be many different collections

of mathematical objects able to fill it. For the reasons given in §2, this forces us to

stipulate that, as a matter of brute fact, the proposed dependence selects for a par-

ticular mathematical relatum from among the eligible candidates. This generates

the brute regularities and empirical access problems, and so mounting this horn

destroys our understanding of mathematical explanation.

The first horn vindicates PREMISE 1, the second horn vindicates PREMISE 2,

and our choice of horns seems to be a forced decision between exhaustive and mu-

tually exclusive alternatives. Along with the three confirming instances identified

in §§2–4, this provides considerable (defeasible) support for CONCLUSION.

6. The Siren Call of Structuralism

One might worry that my dilemma has limited scope. Perhaps the second horn

only threatens a traditional form of platonism, on which the entities answering to

27

mathematical theories are abstract objects whose intrinsic natures determine the

mathematical relations they bear to one another. On another well-established form

of platonism, the entities answering to our mathematical theories are abstract struc-

tures that are independent of, and explanatorily prior to, any possible instantiation

of them by any particular system of objects. On this structuralism, a non-empty

mathematical singular term refers to a position in a mathematical structure whose

nature is exhausted by the relations it bears to other positions in the structure.19

At a distance, it appears that adopting structuralism will blunt the second horn

of my dilemma. For a given explanation, if the supposedly different candidates

for mathematical grounds are merely different instantiations of a unique mathem-

atical structure, then we can say that the posited dependence relation selects for the

structure itself, blocking the underdetermination that leads to the brute regularities

and empirical access problems. However, on closer inspection, things are not so

simple. Even if we assume that there is a unique mathematical structure picked out

by each of our non-algebraic mathematical theories (the uniqueness assumption),

we can only block two of the three sources of underdetermination I have identified.

Worse still, there is reason to doubt the uniqueness assumption. If it is false, then

we are no better off in adopting structuralism.

For concreteness, consider the cicada explanation outlined in §3, which appeals

to properties of natural numbers. On the uniqueness assumption, there is a unique

structure answering to our theory of natural numbers: being a sequence closed un-

der a successor function with a unique first element. Natural numbers are positions19The structuralism considered here is developed and defended in detail in Shapiro 1997. There are

many other ways of spelling out the intuitive idea that mathematics is a science of structure. Whileinteresting and worthy of attention, they are not relevant to our discussion. See Reck & Price 2000for an excellent survey.

28

in this structure. When a system of objects instantiates the natural-number struc-

ture, objects in that system ‘occupy’ the natural-number positions. Now, a collec-

tion of positions in a structure can be treated as a system of objects which can itself

instantiate a distinct structure. So, we can consider various different sequences

of positions in the structure described by set theory as instantiating the natural-

number structure. On structuralism, this is precisely what we do when we provide

set-theoretic models of the natural numbers. On this view, the existence of many

different set-theoretic models of the natural numbers does not underdetermine the

interpretation of the mathematics in the cicadas explanation. The one and only cor-

rect interpretation of this mathematics is in terms of the natural-number structure

itself.

A similar move blocks underdetermination in our choice of objects within a

given interpretation. I suggested that, by applying a uniform translation, we end

up with a distinct collection of mathematical objects with which we can run the

explanation equally well. However, we can only do this because the result of a

uniform translation of the natural numbers is a sequence of natural numbers that

instantiates the natural-number structure. In the example I gave, 12 occupies the

1-position, 24 occupies the 2-position, and so on, where the successor function s is

defined as s(x) = x+12. Measuring life-cycles in months does not yield a distinct

candidate for the mathematical ground of the explanandum. The explanatory prop-

erties will be those had by the 13-position and 17-position in the natural-number

structure, under any uniform translation.

Unfortunately, none of this helps with the third source of underdetermination.

We have the choice of appealing to the fact that 13 and 17 are irreducible or the

fact that 13 and 17 are prime. Both can be used to run the explanation equally

29

well, yet these are distinct properties. The uniqueness assumption doesn’t help.

Nor should we think this is a special case. We saw in §2 that there are different

properties we might appeal to in running the explanation of why soap formations

satisfy Plateau’s laws. More generally, for a given physical phenomenon, it seems

naıve to assume there is a unique mathematical property in terms of which it is

best modelled and explained. So, there remains a source of underdetermination on

which to hang my arguments from §2, keeping the second horn of my dilemma

sharp. One might nevertheless see this as progress. Structuralism has blocked two

out of three sources of underdetermination. Perhaps with some further ingenuity,

we can block the third. But such optimism is premature.

We have seen that two sources of underdetermination can be blocked on the

uniqueness assumption. But the uniqueness assumption is controversial. In fact,

the very ingredients of structuralism that allow us to block underdetermination (on

the uniqueness assumption) provide compelling reasons to doubt it. The ingredi-

ents are the flexible natures of mathematical objects, as both positions in structures

and objects able to occupy those positions. It is natural to individuate structures

on the basis of isomorphism, so that isomorphic structures are identical, and non-

isomorphic structures are distinct. However, if we can make sense of a position of

one structure occupying a position of another structure, then we can make sense of

a permutation of the positions of a given structure. For example, take the natural-

number structure and permute the first and second position. We end up with a

new structure that is isomorphic to the original. In this way, it seems, there are

infinitely-many distinct yet isomorphic structures equally eligible for being the

natural-number structure. In light of arguments like this, Stewart Shapiro, a key

defender of the present form of structuralism, concedes that ‘an ontological realist

30

cannot simply stipulate that there is at most one structure for each isomorphism

type’ (2006: 143).20

Structuralism, combined with an account of mathematical explanation that

posits physical-on-mathematical dependence, faces underdetermination in which

mathematical property the posited dependence selects for, and which of the infin-

itely many structures answering to the relevant mathematical theory is selected for.

These sources of underdetermination are independent and cross-cutting. Follow-

ing my arguments in §2, this combination of views faces the brute regularities and

empirical access problems, so adopting structuralism fails to blunt the second horn

of my dilemma.

7. Conclusions

I do not want to overreach. I have not shown that EIA fails. Nor have I shown that

it is impossible to account for mathematical explanation in a way that supports pla-

tonism. I have provided considerable (albeit defeasible) support for the following

claim.

CONCLUSION

Any account of mathematical explanation apt to bolster EIA fails to

provide an adequate understanding of mathematical explanation.

In doing so, I have dispelled the appearance that the balance of evidence currently

favours EIA. As things now stand, the balance of evidence counts against EIA. I

mentioned in §1 that there are accounts of mathematical explanation that are prom-

ising for undermining EIA. Robert Knowles and Juha Saatsi (2019) provide an20See Assadian 2018 for more on this and related problems for structuralism.

31

account of mathematical explanation according to which they explain by locat-

ing physical facts on which their explananda counterfactually depend. Mary Leng

(2012) provides an account according to which mathematical explanations identify

structural features of the physical system in virtue of which their explananda had

to occur. And, in light of our discussion in §4, we can add Lyon’s (2012) causal

relevance account to this list.

Perhaps these are genuine rivals. Perhaps they offer different but compatible

theoretical perspectives on the same phenomenon. Either way, they are compatible

with the view that the mathematics in mathematical explanations merely serves to

represent otherwise elusive explanatory physical facts. As yet, there is no reason

to think they run into difficulties that undermine their capacity to provide genuine

understanding of how mathematical explanations work. So, on balance, we have

reason to side with the critics of EIA.

Acknowledgements

I gratefully acknowledge funding from the British Academy, without which this

paper would not have been possible. Thanks to Scott Shalkowski, Jess Isserow,

Simon Hewitt, Will Gamester, Juha Saatsi, Seamus Bradley, David Liggins, Chris

Daly, and two anonymous referees from this journal. Their comments on various

iterations of this paper were invaluable. I am grateful to Christopher Pincock for

enlightening discussion. Thanks also to audiences at the University of Cambridge,

the University of Manchester, the University of Chieti-Pescara, and the University

of Santiago de Compostela for their insightful questions. Special thanks to Emily

Shercliff for patient discussion and unwavering support.

32

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